Interpretation of Inaccessible Sets in Martin-L\"of Type Theory with One Mahlo Universe

Abstract

Rathjen proved that Aczel's constructive set theory $\mathbf{CZF}$ extended with inaccessible sets of all transfinite orders can be interpreted in Martin-L\"{o}f type theory $\mathbf{MLTT}$ extended with Setzer's Mahlo universe and another universe above it. In this paper we show that this interpretation can be carried out bottom-up without the universe above the Mahlo universe, provided we add an accessibility predicate instead. If we work in Martin-L\"{o}f type theory with extensional identity types the accessibility predicate can be defined in terms of $\mathrm{W}$-types. The main part of our interpretation has been formalised in the proof assistant Agda.